Properties

Label 8450.2.a.n
Level $8450$
Weight $2$
Character orbit 8450.a
Self dual yes
Analytic conductor $67.474$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 8450 = 2 \cdot 5^{2} \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8450.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(67.4735897080\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 130)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q + q^{2} - 2 q^{3} + q^{4} - 2 q^{6} - 4 q^{7} + q^{8} + q^{9} + O(q^{10}) \) \( q + q^{2} - 2 q^{3} + q^{4} - 2 q^{6} - 4 q^{7} + q^{8} + q^{9} + 2 q^{11} - 2 q^{12} - 4 q^{14} + q^{16} - 2 q^{17} + q^{18} - 6 q^{19} + 8 q^{21} + 2 q^{22} - 6 q^{23} - 2 q^{24} + 4 q^{27} - 4 q^{28} + 2 q^{29} + 6 q^{31} + q^{32} - 4 q^{33} - 2 q^{34} + q^{36} - 2 q^{37} - 6 q^{38} - 10 q^{41} + 8 q^{42} + 10 q^{43} + 2 q^{44} - 6 q^{46} - 12 q^{47} - 2 q^{48} + 9 q^{49} + 4 q^{51} - 2 q^{53} + 4 q^{54} - 4 q^{56} + 12 q^{57} + 2 q^{58} - 10 q^{59} + 2 q^{61} + 6 q^{62} - 4 q^{63} + q^{64} - 4 q^{66} - 12 q^{67} - 2 q^{68} + 12 q^{69} - 10 q^{71} + q^{72} + 10 q^{73} - 2 q^{74} - 6 q^{76} - 8 q^{77} - 4 q^{79} - 11 q^{81} - 10 q^{82} + 8 q^{84} + 10 q^{86} - 4 q^{87} + 2 q^{88} + 14 q^{89} - 6 q^{92} - 12 q^{93} - 12 q^{94} - 2 q^{96} + 14 q^{97} + 9 q^{98} + 2 q^{99} + O(q^{100}) \)

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
0
1.00000 −2.00000 1.00000 0 −2.00000 −4.00000 1.00000 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(5\) \(1\)
\(13\) \(1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 8450.2.a.n 1
5.b even 2 1 1690.2.a.e 1
13.b even 2 1 650.2.a.c 1
39.d odd 2 1 5850.2.a.cb 1
52.b odd 2 1 5200.2.a.bd 1
65.d even 2 1 130.2.a.c 1
65.g odd 4 2 1690.2.d.e 2
65.h odd 4 2 650.2.b.g 2
65.l even 6 2 1690.2.e.a 2
65.n even 6 2 1690.2.e.g 2
65.s odd 12 4 1690.2.l.a 4
195.e odd 2 1 1170.2.a.d 1
195.s even 4 2 5850.2.e.u 2
260.g odd 2 1 1040.2.a.b 1
455.h odd 2 1 6370.2.a.l 1
520.b odd 2 1 4160.2.a.t 1
520.p even 2 1 4160.2.a.c 1
780.d even 2 1 9360.2.a.by 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
130.2.a.c 1 65.d even 2 1
650.2.a.c 1 13.b even 2 1
650.2.b.g 2 65.h odd 4 2
1040.2.a.b 1 260.g odd 2 1
1170.2.a.d 1 195.e odd 2 1
1690.2.a.e 1 5.b even 2 1
1690.2.d.e 2 65.g odd 4 2
1690.2.e.a 2 65.l even 6 2
1690.2.e.g 2 65.n even 6 2
1690.2.l.a 4 65.s odd 12 4
4160.2.a.c 1 520.p even 2 1
4160.2.a.t 1 520.b odd 2 1
5200.2.a.bd 1 52.b odd 2 1
5850.2.a.cb 1 39.d odd 2 1
5850.2.e.u 2 195.s even 4 2
6370.2.a.l 1 455.h odd 2 1
8450.2.a.n 1 1.a even 1 1 trivial
9360.2.a.by 1 780.d even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8450))\):

\( T_{3} + 2 \)
\( T_{7} + 4 \)
\( T_{11} - 2 \)
\( T_{17} + 2 \)
\( T_{31} - 6 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -1 + T \)
$3$ \( 2 + T \)
$5$ \( T \)
$7$ \( 4 + T \)
$11$ \( -2 + T \)
$13$ \( T \)
$17$ \( 2 + T \)
$19$ \( 6 + T \)
$23$ \( 6 + T \)
$29$ \( -2 + T \)
$31$ \( -6 + T \)
$37$ \( 2 + T \)
$41$ \( 10 + T \)
$43$ \( -10 + T \)
$47$ \( 12 + T \)
$53$ \( 2 + T \)
$59$ \( 10 + T \)
$61$ \( -2 + T \)
$67$ \( 12 + T \)
$71$ \( 10 + T \)
$73$ \( -10 + T \)
$79$ \( 4 + T \)
$83$ \( T \)
$89$ \( -14 + T \)
$97$ \( -14 + T \)
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